Solutions Of Boundary Value Problems For Integro-Differential Equations In BANACH Spaces

With the great development of science and technology, all sorts of nonlinear problems have resulted from physics , chemistry , mathematics , biology , medicine , economics ,cybernetics and so on. These nonlinear problems have brought people's wide attention. At present, nonlinear functional analysis has been one of the most important branch of learning in modern mathematics. It provides an effect theoretical tool for studying many nonlinear problems, mainly including topological degree method , variational method , partial ordering method and analytic method and so on. In 1912, L. E. J. Brouwer established the conception of tppological degree for finite dimensional space. J. Leray and J. Schauder have extended the conception to completely continuous field of Banach space in 1934, afterward E. Rothe, H. Amann [1], M. A. Krasnoselskii [2], K. Deimling [3], L. Nirenberg [4] etc. have carried on embedded research on topological degree theory and cone theory. In China, many well-known mathematicians such as K. C. Chang [5], W. Y. Chen [6], D. J. Guo [7] etc., had profound works in various field of nonlinear functional analysis [8-11, 18-21].Let E be a real Banach space. Consider the Cauchy problemx'= f(t,x),x(t₀) = x₀, (a)in which f ∈ C[[a,b] × D,E], D (?) E, t₀ ∈ [a,b], x₀ ∈ D. If dim E = ∞, in 1950, J. Dieudonne [12] gave an example which indicates the Cauchy problem (a) has no solution in E, see section 1 of Chapter 1. This counterexample indicates the famous Cauchy-Peano theorem may be wrong in the infinite dimensional spaces and there are essential difficulties to investigate the ordinary differential equations in infinite dimensional spaces . Till the end of 80's of last century, with many mathematicians making great efforts, many monographs [13-16] about ordinary differential equations in Banach spaces appeared, which indicated that the theory of ordinary differential equations in Banach spaces had come into being primarily. In infinite dimensional Banach spaces, K. Deimling [13] (Theorem 2.1) generalized the Cauchy-Peano theorem, in which the conditions of compactness type were put forward. Under conditions of dissipative type, K. Deimling [13](Theorem 3.2) gave the Cauchy problem (a) has exactly one solution. By using the method of upper and lower solutions and monotone iterative technique, S. W. Du and V. Lakshmikantham [17] obtained the maximal and minimal solutions of the Cauchy problem (a). By way of complementarity, J. X. Sun and Y. Sun [18] obtained the maximal and minimal generalized solutions of Cauchy problem (a). D. J.Guo [7,9,21] and V. Lakshmikantham [7,21] etc. had devoted themselves to the theory of cone in abstract space and utilized partial ordering method to investigate the differential equations in Banach spaces. On investigating method, by establishing comparison results and using the method of upper and lower solutions, the maximal and minimal solutions for various ordinary differential equations in Banach spaces are obtained, see V. Lakshmikantham, S.Leela and A. S. Vatsala [22], L. H. Erbe and D. J. Guo [23], D. J. Guo [19,20,31-33], J. X. Sun [24,25,26], Z. L. Wei [27], J. G. Wang [28], L. S. Liu [29,30], G. X. Song [34] and X. Y. Liu and C. X. Wu [35] etc.. In this situation, the function on the right-hand side of the equation is required to be nondecreasing and the equation must has upper and lower solutions. But lots of problems don't satisfy this, especially the function is singular, which implies that the function is usually decreasing. On the other hand, the upper and lower solutions of lots of equations are difficult to find, or don't exist at all. Moreover, usually, this method can't obtain multiple solutions of problems.In this thesis, by using the theory of topological degree and the theory of Kuratowski measures noncompactness, under the function only satisfying generic conditions, we investigate the existence of solution and multiple solutions for boundary value problems of higher order integro-differential equations, boundary value problems of nonlinear singular integro-differential equations and boundary value problems of nonlinear impulsive integro-differential equations in Banach spaces. Finally, we obtain some new fixed point theorems and generalize the Acute angle principle and the Altman's fixed point theorem.This thesis is composed of four chapters.Chapter 1 investigates the boundary value problems of higher order nonlinear integro-differential equations in Banach spaces. In paper [32], D. J. Guo obtained the existence of maximal and minimal solutions for the initial value problem of a class of integro-differential equations of Volterra type by establishing a comparison result. In § 1.2, at first, we obtain a new Taylor formula and transform the boundary value problems of a class of nonlinear integro-differential equations in Banach spaces into an integral equation by it. Then, by using the Schauder fixed point theorem, the existence of solutions for the boundary value problems of a class of nonlinear integro-differential equations in Banach spaces is obtained. Because our nonlinear term obtains from u to vf-n^l\ the space of our studying is Cn¹[J, J5](J = [0,a], a > 0), and it is difficult to judge the relative compactness of a bounded set of C""1^, E] . In paper [34], by using some new differential and integral inequalities, the unique solution and iterative approximation of the two-point boundary value problem for integro-differential equations in Banach spaces are investigated. However, the nonlinear term doesn't obtain v! and Su, and it is impossible to obtain multiple solutions. In § 1.3, the existence of multiple solutions for a boundary value problem of a class of nonlinear integro-differential equations of mixed type in Banach spaces is obtained by using fixed point index theory. To obtain multiple solutions of boundary value problems, a cone P is introduced, and the open set of Cn¹[J, P] is difficult to be constructed. The main results of this Chapter have been published on Journal of Shangdong University and Journal of Acta Anal Funct. Appl..In Chapter 2, we obtain the existence of solutions for the following boundary value problem of singular nonlinear integro-differential equation of mixed type in real Banach space E :where 6 denotes the zero element of E,(Tu)(t) = f k(t,s)u(s)ds, (Su)(t) = f h(t,s)u(s)ds,Vt € J,JO JOk G C[D,R+](D = {(t,s) € J xj :t> s}),h € C[J x J,R+},. here J = [0,1], and R+ denotes the set of all nonnegative numbers, the nonlinear term f{t,vb,vi,--- ,vn+i) may be singular at t = 0,t = l,v{ = 0(i = 0,1,-â– â–  ,n-l).Singular nonlinear two-point boundary value problems of the ordinary differential equations appear frequently in applications. With Taliaferro [36] treating the general problem, Callegari and Nachman [37] considered existence questions in boundary layer theory, and Luning and Perry [38] obtained constructive results for generalized Emden-Fbwler problems. Results have also been obtained for singular boundary value problems arising in reaction-diffusion theory and in non-Newtonian fluid theory [39]. Singular nonlinear two-point boundary value problems of the ordinary differential equations have made great progress in recent years [40-57]. Because of the difficulties of compactness and continuity arising from singularity, usually in the literature to establish the existence of a solution to a singular problem an approximate family( one approximate problem for each n € {1,2,???} ) of nonsingular problems is discussed at first. The idea is to establish the existence of a solution yn (for each n 6 {1,2, ? ? ?}) to the nonsingular problem, and then a solution to the singular problem is obtained by letting n —> oo in iV(here N is some subsequence of {1,2, ???}). As much as we know, there are a few papers(only [58-59]) to consider the singular problems in Banach spaces. In this chapter, we shall not be concerned with above strategy but instead will construct a new convex closed set and find the connectionuse from ||tz||c to Hu^""1^^ (u belongs to the convex closed set). Schauder's fixed point theorem is used directly to obtain the existence of solutions for (b). When nonlinear term / hasn't u^""1), we use Krasnoselskii's fixed point theorem directly and obtain the existences of multiple solutions for (b). Some main results of this Chapter have been published on Beijing University 2004 Doctoral Forum of China, and Some have been accepted by J. Sys. Sci. and Math. Scis..Chapter 3 investigates the boundary value problems of nonlinear impulsive integro-differential equations in Banach spaces. The theory of impulsive differential equations has been emerging as an important area of investigation in recent years (see [60]). But the corresponding theory for impulsive integro-differential equations in Banach spaces is yet to be developed. It is interested by many mathematical researchers. It has arisen a series of research results about this aspect, see [61-70]. This theory is more plentiful than before theory of differential equations, because all the structure of its emergence has deep physical background and realistic mathematical modeland coincides with many phenomena in nature. Therefore the research on impulsive differential equations is valuable. In §3.2, a boundary value problem of nth order nonlinear impulsive integro-differential equations in Banach spaces is investigated, and by using the Monch fixed point theorem, the existence of solutions of it is obtained. In a general way, to overcome the difficulties arising from singularity, approach or a cone which is constructed on account of the Green function are used. But, this way isn't applicable for boundary value problem of nonlinear singular impulsive integro-differential equations in Banach spaces. In §3.3, by constructing a new cone on account of the characteristic of nonlinear term of the singular problems, and utilizing the cone to construct a new special convex closed set, we triumphantly overcome the singularity and obtain the existence of solutions for boundary value problem of nonlinear singular impulsive integro-differential equations in Banach spaces. In §3.4, an example for infinite system of scalar second order singular impulsive integro-differential equations is offered. The main results of this Chapter have been submitted.In Chapter 4, we generalize the Acute angle principle and the Altman's fixed point theorem, and a new theorem is obtained which has at least both a fixed point and a zero point at the same time. The fixed point theory of nonlinear operators has extensive applications in many fields of mathematics, especially in all sorts of nonlinear differential and integral equations. Many important fixed point theorems are deduced by famous Acute angle principle(see M. A. Krasnoselskii [2]). In §4.2, by the topological degree theory, we generalize the acute angle principle, and an example is offered. In §4.3, we generalize the Altman's fixed point theorem. And a new theorem is obtained which has at least both a fixed point and a zero point at the same time. The main results of this Chapter have been submitted.